Kashaev's conjecture and the Chern-Simons invariants of knots and links

R.M. Kashaev conjectured that the asymptotic behavior of his link invariant, which equals the colored Jones polynomial evaluated at a root of unity, determines the hyperbolic volume of any hyperbolic link complement. We observe numerically that for knots $6_3$, $8_9$ and $8_{20}$ and for the Whitehead link, the colored Jones polynomials are related to the hyperbolic volumes and the Chern-Simons invariants and propose a complexification of Kashaev's conjecture.